Breaching the 2-Approximation Barrier for Connectivity Augmentation: A Reduction to Steiner Tree

The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem ( \(\mathrm{CAP}\) ) is arguably one of the most basic problems in this area: given a \(k\) (-edge)-connected graph \(G\) and a set of extra edges (links), select a minimum cardinality subset \(A\) of links such that adding \(A\) to \(G\) increases its edge connectivity to \(k+1\) . Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and Jájá, SIAM J. Comput., 10 (1981), pp. 270–283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290–306] that \(\mathrm{CAP}\) can be reduced to the case \(k=1\) , also known as the tree augmentation problem ( \(\mathrm{TAP}\) ) for odd \(k\) , and to the case \(k=2\) , also known as the cactus augmentation problem ( \(\mathrm{CacAP}\) ) for even \(k\) . Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC’20, ACM, New York, 2020, pp. 815–825], several better than 2 approximation algorithms were known for \(\mathrm{TAP}\) , culminating with a recent \(1.458\) approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC’18, ACM, New York, 1918, pp. 632–645]. However, for \(\mathrm{CacAP}\) the best known approximation was 2. In this paper we breach the 2 approximation barrier for \(\mathrm{CacAP}\) , hence, for \(\mathrm{CAP}\) , by presenting a polynomial-time \(2\ln (4)-\frac{967}{1120}+\varepsilon \lt 1.91\) approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for \(\mathrm{TAP}\) either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al., ICALP ’14, Springer, Berlin, 2014, pp. 800–811]. This reduction is not approximation preserving, and using the current best approximation factor for a Steiner tree [Byrka et al., J. ACM, 60 (2013), 6] as a black box would not be good enough to improve on 2. To achieve the latter goal, we “open the box” and exploit the specific properties of the instances of a Steiner tree arising from \(\mathrm{CacAP}\) . In our opinion this connection between approximation algorithms for survivable network design and Steiner-type problems is interesting, and might lead to other results in the area.